Short Distance Asymptotics for a Generalized Two-point Scaling Function in the Two-dimensional Ising Model

  • Thomas Bothner
  • William Warner
Open Access


In the 1977 paper of McCoy et al. (J. Math. Phys. 18, 1058–1092, 1977) it was shown that the limiting two-point correlation function in the two-dimensional Ising model is related to a second order nonlinear Painlevé function. This result identified the scaling function as a tau-function and the corresponding connection problem was solved by Tracy (Commun. Math. Phys. 142, 297–311, 1991), see also the works by Tracy and Widom (Commun. Math. Phys. 190, 697–721, 1998). Here we present the solution to a certain generalized version of the above connection problem which is obtained through a refinement of the techniques chosen in Bothner (J. Stat. Phys. 170, 672–683, 2018).


Ising model Generalized 2-point function Short distance expansion 

Mathematics Subject Classification (2010)

Primary 82B20 Secondary 70S05 34M55 


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College London, StrandLondonUK
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

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